\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  45.0 & -6.00 x_{1} & -1.00 x_{2} & -3.00 x_{3} & -9.00 x_{4} & -3.00 x_{5} & -3.00 x_{6} & -2.00 x_{7} & +  4.00 x_{8}\\
 x_{10}   &  -1.0 & +  8.00 x_{1} & -2.00 x_{2} & -2.00 x_{3} & +  4.00 x_{4} & +  1.00 x_{5} & -1.00 x_{6} & -3.00 x_{7} & -10.00 x_{8}\\
 x_{11}   &  12.0 & -9.00 x_{1} & -9.00 x_{2} & +  1.00 x_{3} & +  3.00 x_{4} & +  6.00 x_{5} & -9.00 x_{6} & +  2.00 x_{7} & +  4.00 x_{8}\\
 x_{12}   &  12.0 & -7.00 x_{1} & +  2.00 x_{2} & -4.00 x_{3} & +  2.00 x_{4} & -4.00 x_{5} & -4.00 x_{6} & +  2.00 x_{7} & -2.00 x_{8}\\
 x_{13}   &  1.0 & + 10.00 x_{1} &   & +  8.00 x_{3} & +  4.00 x_{4} & -8.00 x_{5} & -6.00 x_{6} & +  3.00 x_{7} & -3.00 x_{8}\\
 x_{14}   &  -18.0 & +  3.00 x_{1} & + 10.00 x_{2} & -3.00 x_{3} &   & -2.00 x_{5} & +  5.00 x_{6} &   & -3.00 x_{8}\\
 x_{15}   &  -46.0 & -7.00 x_{1} & +  3.00 x_{2} & +  8.00 x_{3} & +  8.00 x_{4} & +  4.00 x_{5} & -9.00 x_{6} & +  1.00 x_{7} & +  4.00 x_{8}\\
\hline
z    &  0.0 & +  1.00 x_{1} & -2.00 x_{2} & +  2.00 x_{3} & +  4.00 x_{4} &   & +  2.00 x_{6} & -3.00 x_{7} & +  4.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,15$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  6.00 y_{9} & -8.00 y_{10} & +  9.00 y_{11} & +  7.00 y_{12} & -10.00 y_{13} & -3.00 y_{14} & +  7.00 y_{15}\\
 y_{2}   &  1.0 & +  1.00 y_{9} & +  2.00 y_{10} & +  9.00 y_{11} & -2.00 y_{12} &   & -10.00 y_{14} & -3.00 y_{15}\\
 y_{3}   &  1.0 & +  3.00 y_{9} & +  2.00 y_{10} & -1.00 y_{11} & +  4.00 y_{12} & -8.00 y_{13} & +  3.00 y_{14} & -8.00 y_{15}\\
 y_{4}   &  1.0 & +  9.00 y_{9} & -4.00 y_{10} & -3.00 y_{11} & -2.00 y_{12} & -4.00 y_{13} &   & -8.00 y_{15}\\
 y_{5}   &  1.0 & +  3.00 y_{9} & -1.00 y_{10} & -6.00 y_{11} & +  4.00 y_{12} & +  8.00 y_{13} & +  2.00 y_{14} & -4.00 y_{15}\\
 y_{6}   &  1.0 & +  3.00 y_{9} & +  1.00 y_{10} & +  9.00 y_{11} & +  4.00 y_{12} & +  6.00 y_{13} & -5.00 y_{14} & +  9.00 y_{15}\\
 y_{7}   &  1.0 & +  2.00 y_{9} & +  3.00 y_{10} & -2.00 y_{11} & -2.00 y_{12} & -3.00 y_{13} &   & -1.00 y_{15}\\
 y_{8}   &  1.0 & -4.00 y_{9} & + 10.00 y_{10} & -4.00 y_{11} & +  2.00 y_{12} & +  3.00 y_{13} & +  3.00 y_{14} & -4.00 y_{15}\\
\hline
z    &  -0 & -45.00 y_{9} & +  1.00 y_{10} & -12.00 y_{11} & -12.00 y_{12} & -1.00 y_{13} & + 18.00 y_{14} & + 46.00 y_{15}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 8 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{15}   &  0.156424581006 & -0.17 y_{3} & +  0.47 y_{11} & +  0.06 y_{4} & +  0.69 y_{12} & -1.09 y_{13} & -0.05 y_{2} & +  0.68 y_{10}\\
 y_{14}   &  0.0558659217877 & +  0.04 y_{3} & +  0.83 y_{11} & -0.00 y_{4} & -0.32 y_{12} & +  0.28 y_{13} & -0.09 y_{2} & +  0.10 y_{10}\\
 y_{9}   &  0.0279329608939 & -0.15 y_{3} & +  0.75 y_{11} & +  0.17 y_{4} & +  0.84 y_{12} & -0.53 y_{13} & -0.04 y_{2} & +  1.05 y_{10}\\
 y_{5}   &  0.569832402235 & +  0.29 y_{3} & -3.96 y_{11} & +  0.25 y_{4} & +  3.09 y_{12} & + 11.34 y_{13} & -0.11 y_{2} & -0.37 y_{10}\\
 y_{1}   &  2.09497206704 & -2.17 y_{3} & + 14.28 y_{11} & +  1.43 y_{4} & + 17.85 y_{12} & -21.66 y_{13} & -0.35 y_{2} & +  2.77 y_{10}\\
 y_{6}   &  2.21229050279 & -2.13 y_{3} & + 11.30 y_{11} & +  1.06 y_{4} & + 14.37 y_{12} & -6.82 y_{13} & -0.14 y_{2} & +  9.78 y_{10}\\
 y_{7}   &  0.899441340782 & -0.13 y_{3} & -0.97 y_{11} & +  0.27 y_{4} & -1.02 y_{12} & -2.96 y_{13} & -0.04 y_{2} & +  4.42 y_{10}\\
 y_{8}   &  0.430167597765 & +  1.37 y_{3} & -6.38 y_{11} & -0.91 y_{4} & -5.09 y_{12} & + 10.32 y_{13} & +  0.11 y_{2} & +  3.37 y_{10}\\
\hline
z    &  6.94413407821 & -0.37 y_{3} & -9.17 y_{11} & -4.66 y_{4} & -23.68 y_{12} & -22.61 y_{13} & -1.91 y_{2} & -13.10 y_{10}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.368715083799 & +  0.17 x_{15} & -0.04 x_{14} & +  0.15 x_{9} & -0.29 x_{5} & +  2.17 x_{1} & +  2.13 x_{6} & +  0.13 x_{7} & -1.37 x_{8}\\
 x_{11}   &  9.16759776536 & -0.47 x_{15} & -0.83 x_{14} & -0.75 x_{9} & +  3.96 x_{5} & -14.28 x_{1} & -11.30 x_{6} & +  0.97 x_{7} & +  6.38 x_{8}\\
 x_{4}   &  4.66480446927 & -0.06 x_{15} & +  0.00 x_{14} & -0.17 x_{9} & -0.25 x_{5} & -1.43 x_{1} & -1.06 x_{6} & -0.27 x_{7} & +  0.91 x_{8}\\
 x_{12}   &  23.6759776536 & -0.69 x_{15} & +  0.32 x_{14} & -0.84 x_{9} & -3.09 x_{5} & -17.85 x_{1} & -14.37 x_{6} & +  1.02 x_{7} & +  5.09 x_{8}\\
 x_{13}   &  22.6089385475 & +  1.09 x_{15} & -0.28 x_{14} & +  0.53 x_{9} & -11.34 x_{5} & + 21.66 x_{1} & +  6.82 x_{6} & +  2.96 x_{7} & -10.32 x_{8}\\
 x_{2}   &  1.91061452514 & +  0.05 x_{15} & +  0.09 x_{14} & +  0.04 x_{9} & +  0.11 x_{5} & +  0.35 x_{1} & +  0.14 x_{6} & +  0.04 x_{7} & -0.11 x_{8}\\
 x_{10}   &  13.1005586592 & -0.68 x_{15} & -0.10 x_{14} & -1.05 x_{9} & +  0.37 x_{5} & -2.77 x_{1} & -9.78 x_{6} & -4.42 x_{7} & -3.37 x_{8}\\
\hline
z    &  -6.94413407821 & -0.16 x_{15} & -0.06 x_{14} & -0.03 x_{9} & -0.57 x_{5} & -2.09 x_{1} & -2.21 x_{6} & -0.90 x_{7} & -0.43 x_{8}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.368715083799 & +  0.17 x_{15} & -0.04 x_{14} & +  0.15 x_{9} & -0.29 x_{5} & +  2.17 x_{1} & +  2.13 x_{6} & +  0.13 x_{7} & -1.37 x_{8}\\
 x_{11}   &  9.16759776536 & -0.47 x_{15} & -0.83 x_{14} & -0.75 x_{9} & +  3.96 x_{5} & -14.28 x_{1} & -11.30 x_{6} & +  0.97 x_{7} & +  6.38 x_{8}\\
 x_{4}   &  4.66480446927 & -0.06 x_{15} & +  0.00 x_{14} & -0.17 x_{9} & -0.25 x_{5} & -1.43 x_{1} & -1.06 x_{6} & -0.27 x_{7} & +  0.91 x_{8}\\
 x_{12}   &  23.6759776536 & -0.69 x_{15} & +  0.32 x_{14} & -0.84 x_{9} & -3.09 x_{5} & -17.85 x_{1} & -14.37 x_{6} & +  1.02 x_{7} & +  5.09 x_{8}\\
 x_{13}   &  22.6089385475 & +  1.09 x_{15} & -0.28 x_{14} & +  0.53 x_{9} & -11.34 x_{5} & + 21.66 x_{1} & +  6.82 x_{6} & +  2.96 x_{7} & -10.32 x_{8}\\
 x_{2}   &  1.91061452514 & +  0.05 x_{15} & +  0.09 x_{14} & +  0.04 x_{9} & +  0.11 x_{5} & +  0.35 x_{1} & +  0.14 x_{6} & +  0.04 x_{7} & -0.11 x_{8}\\
 x_{10}   &  13.1005586592 & -0.68 x_{15} & -0.10 x_{14} & -1.05 x_{9} & +  0.37 x_{5} & -2.77 x_{1} & -9.78 x_{6} & -4.42 x_{7} & -3.37 x_{8}\\
\hline
z    &  15.5754189944 & -0.01 x_{15} & -0.24 x_{14} & -0.45 x_{9} & -1.80 x_{5} & -1.08 x_{1} & +  1.75 x_{6} & -3.90 x_{7} & +  5.14 x_{8}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.368715083799 & +  0.17 x_{15} & -0.04 x_{14} & +  0.15 x_{9} & -0.29 x_{5} & +  2.17 x_{1} & +  2.13 x_{6} & +  0.13 x_{7} & -1.37 x_{8}\\
 x_{11}   &  9.16759776536 & -0.47 x_{15} & -0.83 x_{14} & -0.75 x_{9} & +  3.96 x_{5} & -14.28 x_{1} & -11.30 x_{6} & +  0.97 x_{7} & +  6.38 x_{8}\\
 x_{4}   &  4.66480446927 & -0.06 x_{15} & +  0.00 x_{14} & -0.17 x_{9} & -0.25 x_{5} & -1.43 x_{1} & -1.06 x_{6} & -0.27 x_{7} & +  0.91 x_{8}\\
 x_{12}   &  23.6759776536 & -0.69 x_{15} & +  0.32 x_{14} & -0.84 x_{9} & -3.09 x_{5} & -17.85 x_{1} & -14.37 x_{6} & +  1.02 x_{7} & +  5.09 x_{8}\\
 x_{13}   &  22.6089385475 & +  1.09 x_{15} & -0.28 x_{14} & +  0.53 x_{9} & -11.34 x_{5} & + 21.66 x_{1} & +  6.82 x_{6} & +  2.96 x_{7} & -10.32 x_{8}\\
 x_{2}   &  1.91061452514 & +  0.05 x_{15} & +  0.09 x_{14} & +  0.04 x_{9} & +  0.11 x_{5} & +  0.35 x_{1} & +  0.14 x_{6} & +  0.04 x_{7} & -0.11 x_{8}\\
 x_{10}   &  13.1005586592 & -0.68 x_{15} & -0.10 x_{14} & -1.05 x_{9} & +  0.37 x_{5} & -2.77 x_{1} & -9.78 x_{6} & -4.42 x_{7} & -3.37 x_{8}\\
\hline
z    &  15.5754189944 & -0.01 x_{15} & -0.24 x_{14} & -0.45 x_{9} & -1.80 x_{5} & -1.08 x_{1} & +  1.75 x_{6} & -3.90 x_{7} & +  5.14 x_{8}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{11} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  2.09802306425 & +  0.08 x_{15} & -0.19 x_{14} & +  0.01 x_{9} & +  0.45 x_{5} & -0.52 x_{1} & -0.19 x_{11} & +  0.31 x_{7} & -0.17 x_{8}\\
 x_{6}   &  0.811037891269 & -0.04 x_{15} & -0.07 x_{14} & -0.07 x_{9} & +  0.35 x_{5} & -1.26 x_{1} & -0.09 x_{11} & +  0.09 x_{7} & +  0.56 x_{8}\\
 x_{4}   &  3.80543657331 & -0.02 x_{15} & +  0.08 x_{14} & -0.10 x_{9} & -0.62 x_{5} & -0.09 x_{1} & +  0.09 x_{11} & -0.36 x_{7} & +  0.32 x_{8}\\
 x_{12}   &  12.0224052718 & -0.10 x_{15} & +  1.38 x_{14} & +  0.12 x_{9} & -8.13 x_{5} & +  0.31 x_{1} & +  1.27 x_{11} & -0.21 x_{7} & -3.01 x_{8}\\
 x_{13}   &  28.1397034596 & +  0.81 x_{15} & -0.78 x_{14} & +  0.08 x_{9} & -8.96 x_{5} & + 13.05 x_{1} & -0.60 x_{11} & +  3.55 x_{7} & -6.48 x_{8}\\
 x_{2}   &  2.02388797364 & +  0.04 x_{15} & +  0.08 x_{14} & +  0.04 x_{9} & +  0.16 x_{5} & +  0.18 x_{1} & -0.01 x_{11} & +  0.05 x_{7} & -0.03 x_{8}\\
 x_{10}   &  5.16688632619 & -0.28 x_{15} & +  0.62 x_{14} & -0.40 x_{9} & -3.05 x_{5} & +  9.59 x_{1} & +  0.87 x_{11} & -5.26 x_{7} & -8.89 x_{8}\\
\hline
z    &  16.992092257 & -0.08 x_{15} & -0.37 x_{14} & -0.57 x_{9} & -1.19 x_{5} & -3.29 x_{1} & -0.15 x_{11} & -3.75 x_{7} & +  6.12 x_{8}\\
\end{array}\]


 $ x_{8} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  1.99942567067 & +  0.08 x_{15} & -0.20 x_{14} & +  0.02 x_{9} & +  0.51 x_{5} & -0.70 x_{1} & -0.21 x_{11} & +  0.41 x_{7} & +  0.02 x_{10}\\
 x_{6}   &  1.13880243071 & -0.06 x_{15} & -0.03 x_{14} & -0.09 x_{9} & +  0.16 x_{5} & -0.66 x_{1} & -0.03 x_{11} & -0.25 x_{7} & -0.06 x_{10}\\
 x_{4}   &  3.98942122425 & -0.03 x_{15} & +  0.10 x_{14} & -0.11 x_{9} & -0.73 x_{5} & +  0.25 x_{1} & +  0.12 x_{11} & -0.55 x_{7} & -0.04 x_{10}\\
 x_{12}   &  10.2733066548 & -0.00 x_{15} & +  1.17 x_{14} & +  0.25 x_{9} & -7.09 x_{5} & -2.94 x_{1} & +  0.98 x_{11} & +  1.57 x_{7} & +  0.34 x_{10}\\
 x_{13}   &  24.3771120498 & +  1.01 x_{15} & -1.23 x_{14} & +  0.37 x_{9} & -6.74 x_{5} & +  6.06 x_{1} & -1.23 x_{11} & +  7.38 x_{7} & +  0.73 x_{10}\\
 x_{2}   &  2.00474284867 & +  0.05 x_{15} & +  0.08 x_{14} & +  0.04 x_{9} & +  0.17 x_{5} & +  0.14 x_{1} & -0.02 x_{11} & +  0.07 x_{7} & +  0.00 x_{10}\\
 x_{8}   &  0.58105454276 & -0.03 x_{15} & +  0.07 x_{14} & -0.05 x_{9} & -0.34 x_{5} & +  1.08 x_{1} & +  0.10 x_{11} & -0.59 x_{7} & -0.11 x_{10}\\
\hline
z    &  20.5488735734 & -0.27 x_{15} & +  0.06 x_{14} & -0.85 x_{9} & -3.29 x_{5} & +  3.32 x_{1} & +  0.44 x_{11} & -7.37 x_{7} & -0.69 x_{10}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.775274228203 & +  0.15 x_{15} & -0.17 x_{14} & +  0.11 x_{9} & +  0.34 x_{5} & +  1.07 x_{6} & -0.17 x_{11} & +  0.68 x_{7} & +  0.09 x_{10}\\
 x_{1}   &  1.73775867918 & -0.09 x_{15} & -0.05 x_{14} & -0.14 x_{9} & +  0.24 x_{5} & -1.53 x_{6} & -0.05 x_{11} & -0.38 x_{7} & -0.10 x_{10}\\
 x_{4}   &  4.42460137962 & -0.05 x_{15} & +  0.09 x_{14} & -0.14 x_{9} & -0.67 x_{5} & -0.38 x_{6} & +  0.11 x_{11} & -0.64 x_{7} & -0.06 x_{10}\\
 x_{12}   &  5.16872102228 & +  0.26 x_{15} & +  1.33 x_{14} & +  0.66 x_{9} & -7.79 x_{5} & +  4.48 x_{6} & +  1.13 x_{11} & +  2.68 x_{7} & +  0.62 x_{10}\\
 x_{13}   &  34.9119359946 & +  0.47 x_{15} & -1.55 x_{14} & -0.48 x_{9} & -5.29 x_{5} & -9.25 x_{6} & -1.54 x_{11} & +  5.08 x_{7} & +  0.14 x_{10}\\
 x_{2}   &  2.24787967884 & +  0.03 x_{15} & +  0.07 x_{14} & +  0.02 x_{9} & +  0.21 x_{5} & -0.21 x_{6} & -0.02 x_{11} & +  0.02 x_{7} & -0.01 x_{10}\\
 x_{8}   &  2.45541671378 & -0.13 x_{15} & +  0.01 x_{14} & -0.20 x_{9} & -0.09 x_{5} & -1.65 x_{6} & +  0.04 x_{11} & -1.00 x_{7} & -0.22 x_{10}\\
\hline
z    &  26.3126201515 & -0.57 x_{15} & -0.12 x_{14} & -1.31 x_{9} & -2.50 x_{5} & -5.06 x_{6} & +  0.27 x_{11} & -8.62 x_{7} & -1.01 x_{10}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{11}   &  4.58578595318 & +  0.87 x_{15} & -0.99 x_{14} & +  0.67 x_{9} & +  2.02 x_{5} & +  6.36 x_{6} & -5.92 x_{3} & +  4.02 x_{7} & +  0.52 x_{10}\\
 x_{1}   &  1.50284280936 & -0.13 x_{15} & -0.00 x_{14} & -0.17 x_{9} & +  0.14 x_{5} & -1.85 x_{6} & +  0.30 x_{3} & -0.58 x_{7} & -0.12 x_{10}\\
 x_{4}   &  4.93695652174 & +  0.05 x_{15} & -0.02 x_{14} & -0.07 x_{9} & -0.44 x_{5} & +  0.33 x_{6} & -0.66 x_{3} & -0.19 x_{7} & -0.00 x_{10}\\
 x_{12}   &  10.3446488294 & +  1.25 x_{15} & +  0.21 x_{14} & +  1.42 x_{9} & -5.51 x_{5} & + 11.66 x_{6} & -6.68 x_{3} & +  7.21 x_{7} & +  1.21 x_{10}\\
 x_{13}   &  27.831270903 & -0.88 x_{15} & -0.02 x_{14} & -1.52 x_{9} & -8.41 x_{5} & -19.07 x_{6} & +  9.13 x_{3} & -1.13 x_{7} & -0.66 x_{10}\\
 x_{2}   &  2.14364548495 & +  0.01 x_{15} & +  0.09 x_{14} & +  0.00 x_{9} & +  0.16 x_{5} & -0.36 x_{6} & +  0.13 x_{3} & -0.07 x_{7} & -0.02 x_{10}\\
 x_{8}   &  2.6483277592 & -0.09 x_{15} & -0.03 x_{14} & -0.17 x_{9} & -0.00 x_{5} & -1.38 x_{6} & -0.25 x_{3} & -0.83 x_{7} & -0.20 x_{10}\\
\hline
z    &  27.5566889632 & -0.33 x_{15} & -0.39 x_{14} & -1.13 x_{9} & -1.95 x_{5} & -3.34 x_{6} & -1.60 x_{3} & -7.53 x_{7} & -0.87 x_{10}\\
\end{array}\]
Final Dictionary
Solution:  27.5566889632
Num Pivots:  5
\end{document}
